Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{4}\) to sketch the graph of the function. $$f(x)=(x+3)^{4}$$

Step-by-Step Solution

Answer

The graph of function \(f(x)=(x+3)^{4}\) will be a horizontal shift to the left by 3 units of the graph of base function \(y=x^{4}\). The function's minimum value will now be at \(x=-3\), and the curve will retain the same shape as the base graph, being symmetric with respect to the y-axis, and increasing rapidly as |x| increases.

Step 1: Understanding the base graph \(y=x^{4}\)

This is the graph of a polynomial function of degree 4. It has an even degree, so there is no horizontal asymptote. This means that the graph increases infinitely in both the positive and negative direction as |x| increases. When \(x=0\), \(y=0\); and as |x| increases, \(y=x^{4}\) also increases, though at a much faster rate. This graph is symmetric with respect to the y-axis because it is an even function.

Step 2: Understand the transformation

The graph of \(f(x)=(x+3)^{4}\) is obtained from the base graph \(y=x^{4}\) by shifting every point 3 units to the left. This is because the +3 is added to the input value or arguments of the function, which results in a horizontal transformation or shift. Therefore, the point (a, a^{4}) on the graph of \(y=x^{4}\) will correspond to the point (a-3, (a-3)^{4}) on the graph of \(f(x)=(x+3)^{4}\).

Step 3: Sketch the graph

Retain the shape of the base graph. Start at the point where \(x=-3\) because that's where the minimum value of the function will now be (due to the left shift). From there, sketch the curve of the function using the same shape as the base graph \(y=x^{4}\), remembering that it is symmetric with respect to the y-axis and increases rapidly as |x| increases.

Key Concepts

Horizontal Shift

Understanding the concept of a horizontal shift is key when graphing transformed polynomial functions. Imagine sliding a graph along the x-axis without altering its shape. This is what we call a horizontal shift.

For the function f(x) = (x + 3)^4, each point of the graph y = x^4 moves 3 units to the left because of the +3 inside the brackets. It’s like every x-value in the original graph is substituted with an x-value that is 3 less. For example, the origin point (0,0) on the graph of y = x^4 shifts to (-3,0) on f(x).

This shift does not affect the shape or orientation of the graph—only its position on the coordinate plane. It's crucial to recognize the +3 as not increasing the value of x, but actually shifting the graph in the opposite direction, due to the 'negative' association in the function's transformation.

Even Degree Polynomials

Polynomials with even degrees exhibit particular characteristics that are useful to note when graphing. An even degree polynomial function, such as y = x^4, typically has a graph that shows symmetry about the y-axis and has no horizontal asymptote. This means that the tails of the graph on either end rise or fall off to infinity.

Moreover, the end behavior of even degree polynomial functions is the same on both ends: as x goes towards negative infinity or positive infinity, the y-values will both tend towards positive infinity in this case, due to the positive coefficient of the leading term. This makes the shape of even degree polynomials predictable, which is especially helpful when graphing transformations of these functions.

Symmetry of Graphs

Graphical symmetry can greatly simplify the process of drawing polynomial functions. The original function y = x^4 is inherently symmetrical about the y-axis. This symmetry reflects the function's behavior in that f(-x) = f(x) for all x. Therefore, to graph f(x), one must consider this symmetry.

When the function is transformed into f(x) = (x + 3)^4, it remains symmetrical but around a new vertical line, x = -3. This is due to our horizontal shift. When you find a point that lies on the graph on one side of the axis of symmetry, you’ll know there will be a corresponding point mirrored across the axis on the other side.

Transformation of Functions

The transformation of functions is a broad concept that encompasses shifts, stretches, compressions, and reflections. When a function undergoes a transformation, its graph is altered from its original form.

For our function f(x) = (x + 3)^4, we only consider a horizontal shift, which is just one type of transformation. The +3 within the brackets indicates this specific shift. Understanding transformations in general allows us to predict the resulting graph without plotting numerous points. It is about visualizing the manipulation of the entire function’s graph as a singular object. This concept simplifies and speeds up graphing and is persuasive in helping understand complex changes to functions.